Problem: A particle moves along a straight line. Its speed is inversely proportional to the square of the distance, $S$, it has traveled. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{dS}{dt}=\dfrac{k}{S^2}$ (Choice B) B $\dfrac{dS}{dt}=\dfrac{k}{t^2}$ (Choice C) C $S(t)=\dfrac{k}{S^2}$ (Choice D) D $S(t)=\dfrac{k}{t^2}$
Solution: The distance traveled is denoted by $S$. The speed is the rate of change of the distance, which means it's represented by $S'(t)$, or $\dfrac{dS}{dt}$. Saying that the speed is inversely proportional to something means it's equal to some constant $k$ divided by that thing. That thing, in our case, is the square of the distance, $S^2$. In conclusion, the equation that describes this relationship is $\dfrac{dS}{dt}=\dfrac{k}{S^2}$.